Question 1

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$-axis. Find the volume of the resulting solid. Select the correct answer.

Accepted Answer

A)  $\frac { 9 } { \pi ^ { 2 } }$ 
B)  $\frac { 27 } { 8 } \pi ^ { 2 }$ 
C)  $\frac { 3 } { \pi }$ 
D)  $\frac { \pi ^ { 2 } } { 9 }$ 
E)  $\frac { \pi } { 9 }$ 
A)  $\frac { 9 } { \pi ^ { 2 } }$ 
B)  $\frac { 27 } { 8 } \pi ^ { 2 }$ 
C)  $\frac { 3 } { \pi }$ 
D)  $\frac { \pi ^ { 2 } } { 9 }$ 
E)  $\frac { \pi } { 9 }$

Question 2

Evaluate the integral.
$$\int e ^ { t } \sqrt { 25 - e ^ { 2 t } } d t$$

Accepted Answer

The answer of Evaluate the integral.
\[\int e ^ { t...

Question 3

A particle moves on a straight line with velocity function $v ( t ) = \sin \omega t \cos ^ { 3 } a t$. Find its position function $s = f ( t )$ if $f ( 0 ) = 0$.

Accepted Answer

A)  $\frac { \cos ^ { 5 } \omega t + 1 } { 5 \omega }$ 
B)  $\frac { \sin ^ { 2 } \omega t + 1 } { 7 a }$ 
C)  $\frac { \sin ^ { 3 } \omega t - 1 } { 3 \omega }$ 
D)  $\frac { 1 - \cos ^ { 4 } a t } { 4 a }$ 
E)  $\frac { 1 - \cos ^ { 6 } a t + 1 } { 6 a }$ 
A)  $\frac { \cos ^ { 5 } \omega t + 1 } { 5 \omega }$ 
B)  $\frac { \sin ^ { 2 } \omega t + 1 } { 7 a }$ 
C)  $\frac { \sin ^ { 3 } \omega t - 1 } { 3 \omega }$ 
D)  $\frac { 1 - \cos ^ { 4 } a t } { 4 a }$ 
E)  $\frac { 1 - \cos ^ { 6 } a t + 1 } { 6 a }$

Question 4

Find the integral using an appropriate trigonometric substitution.
$$\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$$

Accepted Answer

The answer of Find the integral using an appropriate trigonometric...

Question 5

Evaluate the integral.
$$\int _ { x / 6 } ^ { \pi / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$$

Accepted Answer

A)  $- \frac { 1 } { 5 } ( \ln 2 ) ^ { 2 }$ 
B)  $- \frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$ 
C)  0 
D)  $\frac { 1 } { 6 } ( \ln 3 ) ^ { 2 }$ 
E)  $\frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$ 
A)  $- \frac { 1 } { 5 } ( \ln 2 ) ^ { 2 }$ 
B)  $- \frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$ 
C)  0 
D)  $\frac { 1 } { 6 } ( \ln 3 ) ^ { 2 }$ 
E)  $\frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$

Question 6

Evaluate the integral.
$$\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 0 }...

Question 7

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$-axis. Find the volume of the resulting solid. Select the correct answer.

Accepted Answer

A)  $\frac { 9 } { \pi ^ { 2 } }$ 
B)  $\frac { 27 } { 8 } \pi ^ { 2 }$ 
C)  $\frac { 3 } { \pi }$ 
D)  $\frac { \pi ^ { 2 } } { 9 }$ 
E)  $\frac { \pi } { 9 }$ 
A)  $\frac { 9 } { \pi ^ { 2 } }$ 
B)  $\frac { 27 } { 8 } \pi ^ { 2 }$ 
C)  $\frac { 3 } { \pi }$ 
D)  $\frac { \pi ^ { 2 } } { 9 }$ 
E)  $\frac { \pi } { 9 }$

Question 8

Find the integral.
$$\int \frac { d x } { x ( x - 3 ) }$$

Accepted Answer

The answer of Find the integral.
\[\int \frac { d x...

Question 9

Evaluate the integral.
$ \int_{1}^{\infty} \frac{d x}{x \ln x} $

Accepted Answer

A)  $ \frac{1}{2} $ 
B)  $ -\frac{1}{2} $ 
C)  $ -\frac{1}{4} $ 
D)  $ \frac{1}{4} $ 
E)  divergent 
A)  $ \frac{1}{2} $ 
B)  $ -\frac{1}{2} $ 
C)  $ -\frac{1}{4} $ 
D)  $ \frac{1}{4} $ 
E)  divergent

Question 10

Evaluate the following integral
$$\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x$$

Accepted Answer

The answer of Evaluate the following integral
\[\int _ { 0...

Question 11

Evaluate the integral.
$$\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$$

Accepted Answer

A)  $2.8665$ 
B)  $6.5700$ 
C)  $0.7166$ 
D)  $11.8665$ 
E)  None of these 
A)  $2.8665$ 
B)  $6.5700$ 
C)  $0.7166$ 
D)  $11.8665$ 
E)  None of these

Question 12

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with $n = 12$ and the formula
$$R = \frac { 60 D } { \int _ { 0 } ^ { 24 } C ( t ) d t }$$ to estimate the person's cardiac output, where $D$ is the quantity of dye injected in milligrams, $C ( t )$ is the concentration of the dye in the aorta, and $R$ is measured in liters per minute. Round to one decimal place.
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{t} & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 & 24 \
\hline \boldsymbol{C}(\boldsymbol{t}) & 0 & 0 & 2.6 & 6.3 & 9.7 & 7.5 & 4.5 & 3.5 & 2.2 & 0.6 & 0.3 & 0.1 & 0 \
\hline
\end{array}$

Accepted Answer

The answer of Eight milligrams of a dye are injected...

Question 13

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$$

Accepted Answer

The answer of Evaluate the integral using integration by parts...

Question 14

Find the integral.
$$\int \tan ^ { 2 } x \sec ^ { 6 } x d x$$

Accepted Answer

A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
C)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x - \tan x + C$ 
D)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x - \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
C)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x - \tan x + C$ 
D)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x - \frac { 1 } { 3 } \tan ^ { 3 } x + C$

Question 15

Find the integral.
$\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Accepted Answer

A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 3 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
C)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x - \tan x + C$ 
D)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x - \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 3 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
C)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x - \tan x + C$ 
D)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x - \frac { 1 } { 3 } \tan ^ { 3 } x + C$

Question 16

Use a table of integrals to evaluate the integral. Select the correct answer.
$$\int e ^ { - 7 x } \sin 3 x d x$$

Accepted Answer

A)  $- \frac { 1 } { 10 } e ^ { - 7 x } ( 7 \sin 3 x + 7 \cos 3 x ) + C$ 
B)  $- \frac { 1 } { 10 } e ^ { - 7 x } ( 7 \sin 3 x + 3 \cos 3 x ) + C$ 
C)  $- \frac { 1 } { 58 } e ^ { - 7 x } ( 7 \sin 3 x + 7 \cos 3 x ) + C$ 
D)  $- \frac { 1 } { 58 } e ^ { - 7 x } ( 7 \sin 3 x + 3 \cos 3 x ) + C$ 
A)  $- \frac { 1 } { 10 } e ^ { - 7 x } ( 7 \sin 3 x + 7 \cos 3 x ) + C$ 
B)  $- \frac { 1 } { 10 } e ^ { - 7 x } ( 7 \sin 3 x + 3 \cos 3 x ) + C$ 
C)  $- \frac { 1 } { 58 } e ^ { - 7 x } ( 7 \sin 3 x + 7 \cos 3 x ) + C$ 
D)  $- \frac { 1 } { 58 } e ^ { - 7 x } ( 7 \sin 3 x + 3 \cos 3 x ) + C$

Question 17

Find the integral.
$$\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$$

Accepted Answer

The answer of Find the integral.
\[\int \frac { x ^...

Question 18

Find the integral.
$$\int \tan ^ { 3 } x \sec ^ { 5 } x d x$$

Accepted Answer

The answer of Find the integral.
\[\int \tan ^ { 3...

Question 19

Find the integral using an appropriate trigonometric substitution. Select the correct answer.
$$\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$$

Accepted Answer

A)  $\sqrt { 1 - x ^ { 2 } } + C$ 
B)  $\sqrt { 1 - x } + C$ 
C)  $- \sqrt { 1 - x ^ { 2 } } + C$ 
D)  $- \sqrt { 1 - x } + C$ 
A)  $\sqrt { 1 - x ^ { 2 } } + C$ 
B)  $\sqrt { 1 - x } + C$ 
C)  $- \sqrt { 1 - x ^ { 2 } } + C$ 
D)  $- \sqrt { 1 - x } + C$

Question 20

Find the integral.
$$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$$

Accepted Answer

The answer of Find the integral.
\[\int \frac { x ^...

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$ -axis. Find the volume of the resulting solid. Select the correct answer.

Evaluate the integral. $\int e ^ { t } \sqrt { 25 - e ^ { 2 t } } d t$

A particle moves on a straight line with velocity function $v ( t ) = \sin \omega t \cos ^ { 3 } a t$ . Find its position function $s = f ( t )$ if $f ( 0 ) = 0$ .

Find the integral using an appropriate trigonometric substitution. $\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$

Evaluate the integral. $\int _ { x / 6 } ^ { \pi / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$

Evaluate the integral. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$ -axis. Find the volume of the resulting solid. Select the correct answer.

Find the integral. $\int \frac { d x } { x ( x - 3 ) }$

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x \ln x}$

Evaluate the following integral $\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$ . $\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Use a table of integrals to evaluate the integral. Select the correct answer. $\int e ^ { - 7 x } \sin 3 x d x$

Find the integral. $\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Find the integral. $\int \tan ^ { 3 } x \sec ^ { 5 } x d x$

Find the integral using an appropriate trigonometric substitution. Select the correct answer. $\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$

Find the integral. $\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 7: Techniques of Integration

The region under the curve y=3sin⁡2x,0≤x≤πy = 3 \sin ^ { 2 } x , 0 \leq x \leq \piy=3sin2x,0≤x≤π is rotated about the xxx -axis. Find the volume of the resulting solid. Select the correct answer.

Evaluate the integral. ∫et25−e2tdt\int e ^ { t } \sqrt { 25 - e ^ { 2 t } } d t∫et25−e2t​dt

A particle moves on a straight line with velocity function v(t)=sin⁡ωtcos⁡3atv ( t ) = \sin \omega t \cos ^ { 3 } a tv(t)=sinωtcos3at . Find its position function s=f(t)s = f ( t )s=f(t) if f(0)=0f ( 0 ) = 0f(0)=0 .

Find the integral using an appropriate trigonometric substitution. ∫x1−x2dx\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x∫1−x2​x​dx

Evaluate the integral. ∫x/6π/33ln⁡(tan⁡x)7sin⁡xcos⁡xdx\int _ { x / 6 } ^ { \pi / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x∫x/6π/3​7sinxcosx3ln(tanx)​dx

Evaluate the integral. ∫08(x2+8)e−xdx\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x∫08​(x2+8)e−xdx

The region under the curve y=3sin⁡2x,0≤x≤πy = 3 \sin ^ { 2 } x , 0 \leq x \leq \piy=3sin2x,0≤x≤π is rotated about the xxx -axis. Find the volume of the resulting solid. Select the correct answer.

Find the integral. ∫dxx(x−3)\int \frac { d x } { x ( x - 3 ) }∫x(x−3)dx​

Evaluate the integral. ∫1∞dxxln⁡x \int_{1}^{\infty} \frac{d x}{x \ln x} ∫1∞​xlnxdx​

Evaluate the following integral ∫013ln⁡2xxdx\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x∫01​x​3ln2x​dx

Evaluate the integral. ∫019xsin⁡πxdx\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x∫01​9xsinπxdx

Evaluate the integral using integration by parts with the indicated choices of uuu and dvd vdv . ∫6θcos⁡θdθ,u=6θ,dv=cos⁡θdθ\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta∫6θcosθdθ,u=6θ,dv=cosθdθ

Find the integral. ∫tan⁡2xsec⁡6xdx\int \tan ^ { 2 } x \sec ^ { 6 } x d x∫tan2xsec6xdx

Find the integral. ∫tan⁡2xsec⁡6xdx\int \tan ^ { 2 } x \sec ^ { 6 } x d x∫tan2xsec6xdx

Use a table of integrals to evaluate the integral. Select the correct answer. ∫e−7xsin⁡3xdx\int e ^ { - 7 x } \sin 3 x d x∫e−7xsin3xdx

Find the integral. ∫x2−x+2x3−2x2+xdx\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x∫x3−2x2+xx2−x+2​dx

Find the integral. ∫tan⁡3xsec⁡5xdx\int \tan ^ { 3 } x \sec ^ { 5 } x d x∫tan3xsec5xdx

Find the integral using an appropriate trigonometric substitution. Select the correct answer. ∫x1−x2dx\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x∫1−x2​x​dx

Find the integral. ∫x3−3x2+6x−2x3−2x2+xdx\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x∫x3−2x2+xx3−3x2+6x−2​dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$ -axis. Find the volume of the resulting solid. Select the correct answer.

Evaluate the integral. $\int e ^ { t } \sqrt { 25 - e ^ { 2 t } } d t$

A particle moves on a straight line with velocity function $v ( t ) = \sin \omega t \cos ^ { 3 } a t$ . Find its position function $s = f ( t )$ if $f ( 0 ) = 0$ .

Find the integral using an appropriate trigonometric substitution. $\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$

Evaluate the integral. $\int _ { x / 6 } ^ { \pi / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$

Evaluate the integral. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$ -axis. Find the volume of the resulting solid. Select the correct answer.

Find the integral. $\int \frac { d x } { x ( x - 3 ) }$

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x \ln x}$

Evaluate the following integral $\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$ . $\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Use a table of integrals to evaluate the integral. Select the correct answer. $\int e ^ { - 7 x } \sin 3 x d x$

Find the integral. $\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Find the integral. $\int \tan ^ { 3 } x \sec ^ { 5 } x d x$

Find the integral using an appropriate trigonometric substitution. Select the correct answer. $\int \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$

Find the integral. $\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$